The comments come just weeks after England’s Higher Education Funding Council warned that university finances were under pressure after an “unexpected fall” in admissions.

Nationally, numbers were an average of 2.1 per cent lower than universities’ own forecasts, it emerged. Some 57,000 fewer undergraduates started courses across the country this year.

Speaking at the Girls' Schools Association annual conference in Liverpool, Sir Howard said: “One of the startling unintended consequences is that currently, this year, there are about 11,500 empty places in Russell Group universities.”

This “certainly wasn't the intention” of the reforms, he said.

He said the “downward pressure on A-level grades meant that there wasn't as many [students] around to recruit [and] there was some slacking of demand in certain subjects, mainly humanities and social sciences”.

In 2012, universities were given new powers to charge up to £9,000 in tuition fees – almost three times the previous cap – after a sharp drop in direct state funding.

Universities charging more than £7,500 – including every leading university – had to hand back a certain proportion of fixed student places allocated by the Government, which were then distributed to institutions with the lowest fees.

These "core and margin" reforms had an effect on top institutions, Sir Howard suggested.

In a further change, leading universities were also free to recruit unlimited numbers of students with at least two As and a B at A-level to create more competition in the higher education sector and plug gaps created by the "core and margin" programme.

But Sir Howard pointed out that fewer students achieved top grades at A-level in the summer, leaving some universities struggling to fill places.

Next year, ministers have said that universities will be free to recruit as many students with an A and two Bs as they like.

But Sir Howard said: “We believe, in the Russell Group certainly, we could have a further round of empty places next year."

The Russell Group represents 24 universities including Oxford, Cambridge, University College London and the London School of Economics. It is not known which institutions had vacancies.

Hilary French, head of Central Newcastle High School and president elect of the GSA said: "It seems to be ludicrous.

"Our girls got places, but to think there are places sitting there and there were people who would have taken those places."

She suggested that Russell Group universities should have made their unfilled places available through clearing.

"If it was purely wanting to protect their brand and their name that kept them out of clearing then they have missed out on a lot of really good candidates," she said.